The goal of the proposed work is to build a theoretical framework based on phase resetting curves (PRC) for understanding the basic mechanisms of synchronization and phase locking in networks of biological neurons. These phenomena underlie central pattern generation for the production of repetitive motor activity such as locomotion and respiration. Transiently synchronized assemblies of neurons are hypothesized to underlie many cognitive functions, and synchronization is altered in psychiatric disorders such as schizophrenia. Pathological synchronization is involved in epilepsy and tremor. PRCs measure the change in length of a periodic cycle due to an input perturbation to an isolated neuron, and provide useful information regarding the activity of the neuron within a network context. During the previous progress period, two major approaches were explored as theoretical substrates: 1: the phase resetting curves using a spike (or burst) as a perturbation under the assumption of pulsatile coupling and 2) a functional PRC (FPRC) consisting of repeated stimulation with a similar perturbation applied at a fixed delay after each spike (or burst) initiation. The overall goal of the project is to create a very broad theoretical framework that can be used to analyze the biological basis of coordinated oscillatory activity in the nervous system. Therefore we need to develop and test methods to analyze circuits that include neurons exhibiting spike frequency adaptation, neurons that are not intrinsic oscillators, noisy circuits, and circuits whose parameters cannot be defined in detail for every component neuron. The following objectives will be accomplished in the next project period: we will develop and test PRC-based methods to include neurons that are not endogenous pacemakers, to analyze circuits of adapting neurons, and to estimate the robustness of synchronization in the presence of noise and heterogeneity. We will analyze a biological central pattern generator, the pyloric circuit of the lobster and crab, using the methods that we develop. General principles will emerge that should be applicable to many oscillatory networks in the nervous system. A better understanding of these mechanisms will provide tools to determine how synchronization goes awry in neurological disorders such as epilepsy and tremor, or in psychiatric disorders in which cognition is impaired.